direct product, abelian, monomial
Aliases: C6×C30, SmallGroup(180,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C30 |
| C1 — C6×C30 |
| C1 — C6×C30 |
Generators and relations for C6×C30
G = < a,b | a6=b30=1, ab=ba >
Subgroups: 60, all normal (8 characteristic)
C1, C2, C3, C22, C5, C6, C32, C10, C2×C6, C15, C3×C6, C2×C10, C30, C62, C3×C15, C2×C30, C3×C30, C6×C30
Quotients: C1, C2, C3, C22, C5, C6, C32, C10, C2×C6, C15, C3×C6, C2×C10, C30, C62, C3×C15, C2×C30, C3×C30, C6×C30
►Regular action on 180 points
Generators in S180
(1 125 154 64 114 31)(2 126 155 65 115 32)(3 127 156 66 116 33)(4 128 157 67 117 34)(5 129 158 68 118 35)(6 130 159 69 119 36)(7 131 160 70 120 37)(8 132 161 71 91 38)(9 133 162 72 92 39)(10 134 163 73 93 40)(11 135 164 74 94 41)(12 136 165 75 95 42)(13 137 166 76 96 43)(14 138 167 77 97 44)(15 139 168 78 98 45)(16 140 169 79 99 46)(17 141 170 80 100 47)(18 142 171 81 101 48)(19 143 172 82 102 49)(20 144 173 83 103 50)(21 145 174 84 104 51)(22 146 175 85 105 52)(23 147 176 86 106 53)(24 148 177 87 107 54)(25 149 178 88 108 55)(26 150 179 89 109 56)(27 121 180 90 110 57)(28 122 151 61 111 58)(29 123 152 62 112 59)(30 124 153 63 113 60)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
G:=sub<Sym(180)| (1,125,154,64,114,31)(2,126,155,65,115,32)(3,127,156,66,116,33)(4,128,157,67,117,34)(5,129,158,68,118,35)(6,130,159,69,119,36)(7,131,160,70,120,37)(8,132,161,71,91,38)(9,133,162,72,92,39)(10,134,163,73,93,40)(11,135,164,74,94,41)(12,136,165,75,95,42)(13,137,166,76,96,43)(14,138,167,77,97,44)(15,139,168,78,98,45)(16,140,169,79,99,46)(17,141,170,80,100,47)(18,142,171,81,101,48)(19,143,172,82,102,49)(20,144,173,83,103,50)(21,145,174,84,104,51)(22,146,175,85,105,52)(23,147,176,86,106,53)(24,148,177,87,107,54)(25,149,178,88,108,55)(26,150,179,89,109,56)(27,121,180,90,110,57)(28,122,151,61,111,58)(29,123,152,62,112,59)(30,124,153,63,113,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)>;
G:=Group( (1,125,154,64,114,31)(2,126,155,65,115,32)(3,127,156,66,116,33)(4,128,157,67,117,34)(5,129,158,68,118,35)(6,130,159,69,119,36)(7,131,160,70,120,37)(8,132,161,71,91,38)(9,133,162,72,92,39)(10,134,163,73,93,40)(11,135,164,74,94,41)(12,136,165,75,95,42)(13,137,166,76,96,43)(14,138,167,77,97,44)(15,139,168,78,98,45)(16,140,169,79,99,46)(17,141,170,80,100,47)(18,142,171,81,101,48)(19,143,172,82,102,49)(20,144,173,83,103,50)(21,145,174,84,104,51)(22,146,175,85,105,52)(23,147,176,86,106,53)(24,148,177,87,107,54)(25,149,178,88,108,55)(26,150,179,89,109,56)(27,121,180,90,110,57)(28,122,151,61,111,58)(29,123,152,62,112,59)(30,124,153,63,113,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180) );
G=PermutationGroup([[(1,125,154,64,114,31),(2,126,155,65,115,32),(3,127,156,66,116,33),(4,128,157,67,117,34),(5,129,158,68,118,35),(6,130,159,69,119,36),(7,131,160,70,120,37),(8,132,161,71,91,38),(9,133,162,72,92,39),(10,134,163,73,93,40),(11,135,164,74,94,41),(12,136,165,75,95,42),(13,137,166,76,96,43),(14,138,167,77,97,44),(15,139,168,78,98,45),(16,140,169,79,99,46),(17,141,170,80,100,47),(18,142,171,81,101,48),(19,143,172,82,102,49),(20,144,173,83,103,50),(21,145,174,84,104,51),(22,146,175,85,105,52),(23,147,176,86,106,53),(24,148,177,87,107,54),(25,149,178,88,108,55),(26,150,179,89,109,56),(27,121,180,90,110,57),(28,122,151,61,111,58),(29,123,152,62,112,59),(30,124,153,63,113,60)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)]])
C6×C30 is a maximal subgroup of
C62⋊D5
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 6A | ··· | 6X | 10A | ··· | 10L | 15A | ··· | 15AF | 30A | ··· | 30CR |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C6×C30 | C3×C30 | C2×C30 | C62 | C30 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 8 | 4 | 24 | 12 | 32 | 96 |
Matrix representation of C6×C30 ►in GL2(𝔽31) generated by
| 26 | 0 |
| 0 | 5 |
| 5 | 0 |
| 0 | 15 |
G:=sub<GL(2,GF(31))| [26,0,0,5],[5,0,0,15] >;
C6×C30 in GAP, Magma, Sage, TeX
C_6\times C_{30} % in TeX
G:=Group("C6xC30"); // GroupNames label
G:=SmallGroup(180,37);
// by ID
G=gap.SmallGroup(180,37);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-5]);
// Polycyclic
G:=Group<a,b|a^6=b^30=1,a*b=b*a>;
// generators/relations